Is there an extension of Ito's Lemma where $X_t$ is a semi-martingale and $f:\mathbb{R}^d \rightarrow \mathbb{R}$ is a function which is not smooth?
I've been looking but have not found much, any reference is appraciated.
There are versions available for convex $f$ and for $f\in H^1$. Some places to start are On semimartingale decompositions of convex functions of semimartingales (Carlen and Protter) and On Itô s formula for multidimensional Brownian motion (Follmer and Protter).
Doeblin had nothing to do with this. The extension of Ito formula to convex functions was derived by Hiroshi Tanaka and is called the Tanaka formula. https://en.wikipedia.org/wiki/Tanaka%27s_formula
The non-smooth generalization of Ito's Lemma is called the Ito-Doeblin formula. Doeblin has a fascinating personal history.